Bottom Line:
While SDU effectively minimizes functions with funnel-shaped basins, its application to docking in the rotational and translational space SE(3) is not straightforward due to the geometry of that space.The removal of the center-to-center distance turns out to vastly improve the efficiency of the search, because the five-dimensional space now exhibits a well-behaved energy surface suitable for underestimation.This algorithm explores the free energy surface spanned by encounter complexes that correspond to local free energy minima and shows similarity to the model of macromolecular association that proceeds through a series of collisions.

ABSTRACTSimilarly to protein folding, the association of two proteins is driven by a free energy funnel, determined by favorable interactions in some neighborhood of the native state. We describe a docking method based on stochastic global minimization of funnel-shaped energy functions in the space of rigid body motions (SE(3)) while accounting for flexibility of the interface side chains. The method, called semi-definite programming-based underestimation (SDU), employs a general quadratic function to underestimate a set of local energy minima and uses the resulting underestimator to bias further sampling. While SDU effectively minimizes functions with funnel-shaped basins, its application to docking in the rotational and translational space SE(3) is not straightforward due to the geometry of that space. We introduce a strategy that uses separate independent variables for side-chain optimization, center-to-center distance of the two proteins, and five angular descriptors of the relative orientations of the molecules. The removal of the center-to-center distance turns out to vastly improve the efficiency of the search, because the five-dimensional space now exhibits a well-behaved energy surface suitable for underestimation. This algorithm explores the free energy surface spanned by encounter complexes that correspond to local free energy minima and shows similarity to the model of macromolecular association that proceeds through a series of collisions. Results for standard protein docking benchmarks establish that in this space the free energy landscape is a funnel in a reasonably broad neighborhood of the native state and that the SDU strategy can generate docking predictions with less than 5 A ligand interface C(alpha) root-mean-square deviation while achieving an approximately 20-fold efficiency gain compared to Monte Carlo methods.

pcbi-1000191-g001: Funnel-like function and underestimator at a set of local minima indicated by small squares.

Mentions:
A minimization approach which is specific to funnel-like functions can be based on the concept of underestimation. The existence of a funnel implies that the free energy can be locally underestimated by a convex function (Figure 1). The original free energy function is extremely rugged with a huge number of local minima even in a small region of conformational space. Yet its convex underestimator is much smoother and still captures the overall funnel-like landscape, which provides a handle to free energy minimization. The quality of minimization through underestimation depends on the choice of underestimator functions, the way they are constructed and utilized to locate the global minimum, as well as how structured the free energy funnels are in conformational space. The Convex Global Underestimation (CGU) method [26] employed canonical quadratic functions as underestimators without any cross-terms. In that case the underestimator, based on a set of local minima, can be constructed by solving a Linear Programming (LP) problem. Uniformly distributed samples in the neighborhood of the underestimator's global minimum were then used to bias further sampling. The process was iterated with the set of local minima being updated, and the search region being reduced until certain convergence criteria are satisfied. CGU has been a very promising method with various applications in molecular structure prediction, including protein folding [27] and docking small molecules to proteins [28]. However, its restriction of using canonical quadratic functions limits its success in some cases [29], since the principal axes of the free energy surface are not necessarily aligned with the canonical coordinates. The performance further deteriorates as the dimensionality of the search space increases. We have used theoretical analysis to show and simple test problems to demonstrate that this restriction can lead to incorrect convergence [30].

pcbi-1000191-g001: Funnel-like function and underestimator at a set of local minima indicated by small squares.

Mentions:
A minimization approach which is specific to funnel-like functions can be based on the concept of underestimation. The existence of a funnel implies that the free energy can be locally underestimated by a convex function (Figure 1). The original free energy function is extremely rugged with a huge number of local minima even in a small region of conformational space. Yet its convex underestimator is much smoother and still captures the overall funnel-like landscape, which provides a handle to free energy minimization. The quality of minimization through underestimation depends on the choice of underestimator functions, the way they are constructed and utilized to locate the global minimum, as well as how structured the free energy funnels are in conformational space. The Convex Global Underestimation (CGU) method [26] employed canonical quadratic functions as underestimators without any cross-terms. In that case the underestimator, based on a set of local minima, can be constructed by solving a Linear Programming (LP) problem. Uniformly distributed samples in the neighborhood of the underestimator's global minimum were then used to bias further sampling. The process was iterated with the set of local minima being updated, and the search region being reduced until certain convergence criteria are satisfied. CGU has been a very promising method with various applications in molecular structure prediction, including protein folding [27] and docking small molecules to proteins [28]. However, its restriction of using canonical quadratic functions limits its success in some cases [29], since the principal axes of the free energy surface are not necessarily aligned with the canonical coordinates. The performance further deteriorates as the dimensionality of the search space increases. We have used theoretical analysis to show and simple test problems to demonstrate that this restriction can lead to incorrect convergence [30].

Bottom Line:
While SDU effectively minimizes functions with funnel-shaped basins, its application to docking in the rotational and translational space SE(3) is not straightforward due to the geometry of that space.The removal of the center-to-center distance turns out to vastly improve the efficiency of the search, because the five-dimensional space now exhibits a well-behaved energy surface suitable for underestimation.This algorithm explores the free energy surface spanned by encounter complexes that correspond to local free energy minima and shows similarity to the model of macromolecular association that proceeds through a series of collisions.

ABSTRACTSimilarly to protein folding, the association of two proteins is driven by a free energy funnel, determined by favorable interactions in some neighborhood of the native state. We describe a docking method based on stochastic global minimization of funnel-shaped energy functions in the space of rigid body motions (SE(3)) while accounting for flexibility of the interface side chains. The method, called semi-definite programming-based underestimation (SDU), employs a general quadratic function to underestimate a set of local energy minima and uses the resulting underestimator to bias further sampling. While SDU effectively minimizes functions with funnel-shaped basins, its application to docking in the rotational and translational space SE(3) is not straightforward due to the geometry of that space. We introduce a strategy that uses separate independent variables for side-chain optimization, center-to-center distance of the two proteins, and five angular descriptors of the relative orientations of the molecules. The removal of the center-to-center distance turns out to vastly improve the efficiency of the search, because the five-dimensional space now exhibits a well-behaved energy surface suitable for underestimation. This algorithm explores the free energy surface spanned by encounter complexes that correspond to local free energy minima and shows similarity to the model of macromolecular association that proceeds through a series of collisions. Results for standard protein docking benchmarks establish that in this space the free energy landscape is a funnel in a reasonably broad neighborhood of the native state and that the SDU strategy can generate docking predictions with less than 5 A ligand interface C(alpha) root-mean-square deviation while achieving an approximately 20-fold efficiency gain compared to Monte Carlo methods.